Well a quad core cpu has the potential to be twice as fast as a dual core cpu of same speed, but it's up to the software to utilize the potential, otherwise it's wasted. Multi tasking with 4 cores in use the quad core will be twice as fast, web browsers utilize a separate core per tab open, couple that with listening to the music app while browsing the web, it's easy to see the advantages of quad core performance in multi app use. But a single app can also use all the cores if designed to (web browser, some games etc).
Android performance in general can also benefit from quad core. Just go into your phones running apps menu, usually a whole bunch running in the back round, even the keyboard is a separate app that can use a separate core, there are quite a few apps that pile up, quad core will handle normal use better than a dual core, and when pushed can be twice as fast.
I personally went from a dual core phone to a quad core, it was night and day difference to me, but obviously there were other factors involved in the performance gap, like android version.